CM \(\frac{1}{\sqrt{1}}+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+....+\frac{1}{\sqrt{n}}>\sqrt{n}\)
1-a,\(A=\frac{1}{1+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+\frac{1}{\sqrt{3}+\sqrt{4}}+...+\frac{1}{\sqrt{n-1}+\sqrt{n}}\)
b,\(B=\frac{1}{2+\sqrt{2}}+\frac{1}{3\sqrt{2}+2\sqrt{3}}+\frac{1}{4\sqrt{3}+3\sqrt{4}}+...+\frac{1}{100\sqrt{99}+99\sqrt{100}}\)
Tính P = \(\frac{4+\sqrt{3}}{\sqrt{1}+\sqrt{3}}+\frac{8+\sqrt{15}}{\sqrt{3}+\sqrt{5}}+...+\frac{2n+\sqrt{n^2-1}}{\sqrt{n-1}+\sqrt{n+1}}+...+\frac{240+\sqrt{14399}}{\sqrt{119}+\sqrt{121}}\)
Rút gọn các biểu thức,
a> A= \(\frac{1}{\sqrt{1}+\sqrt{2}}\) + \(\frac{1}{\sqrt{2}+\sqrt{3}}\)+ \(\frac{1}{\sqrt{3}+\sqrt{4}}\)+ ......... + \(\frac{1}{\sqrt{n-1}+\sqrt{n}}\)
b> B= \(\frac{1}{\sqrt{1}-\sqrt{2}}\)- \(\frac{1}{\sqrt{2}-\sqrt{3}}\)- \(\frac{1}{\sqrt{3}-\sqrt{4}}\)- .......... - \(\frac{1}{\sqrt{24}-\sqrt{25}}\)
CMR: \(\frac{1}{2\sqrt[3]{1}}+\frac{1}{3\sqrt[3]{2}}+\frac{1}{4\sqrt[3]{3}}+...+\frac{1}{\left(n+1\right)\sqrt[3]{n}}\) với mọi \(n\varepsilonℕ^∗\)
CMR : với mọi số tự nhiên n > 1, ta có :
a) \(\frac{1}{n+1}+\frac{1}{n+2}+...+\frac{1}{n+n}< \frac{3}{4}\)
b) \(1+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+...+\frac{1}{\sqrt{n}}>2\left(\sqrt{n+1}-1\right)\)
tìm phân nguyên của a với
\(A=\sqrt{2}+\sqrt[3]{\frac{3}{2}}+\sqrt[4]{\frac{4}{3}}+...+\sqrt[n+1]{\frac{n+1}{n}}\)
Cho n thuộc N*
Cminh \(\frac{1}{2\sqrt{1}}+\frac{1}{3\sqrt{2}}+....+\frac{1}{\left(n+1\right)\sqrt{n}}< 2\)
Tìm phần nguyên của a , với
a = \(\sqrt{2}+\sqrt[3]{\frac{3}{2}}+\sqrt[4]{\frac{4}{3}}+...+\sqrt[n+1]{\frac{n+1}{n}}\)